# Questions tagged [linear-regression]

For questions about linear regressions, an approach for modeling the relationship between a scalar dependent variable y and one or more explanatory variables.

608 questions
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### Independence between error and regressor

Let the following classical linear regression: $$y_i = x_i \theta + u_i, \quad E(u_i|x_i) \sim N(0, \sigma^2)$$ Can I conclude that $x$ and $u$ are independent? I would like this because I want to ...
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### Forming ANOVA table without observed values.

Need some help in forming the ANOVA table without observed values. I only managed to find SST with the first line of hint, but do not know how to proceed to find SSTreatment or SSError. Any hint/...
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### Derive the expression for $\operatorname{Var}(b_1)$ for the no-intercept linear model y = $b_1$*$x_i$ + $ε_i$

Note: $b_1$ is the estimate for $β_1$ What I tried: $\operatorname{var}(b_1) = \operatorname{var}( Σ(x_iy_i)/Σx_i^2)$ $\operatorname{var}(b_1)= (1/(Σx_i^2)^2) * Σx_i * \operatorname{var}(y_i)$ ...
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### How to determine where the data is not linear anymore

I have some data. When I plot these data, the graph looks very linear when $x>x_0$ (In the following figure $x_0$ is shown by dashed lines). One way to determine the exact location where the graph ...
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### Spurious Regression and Co-integration

I downloaded a ppt file from Spurious Regression and Co-integration On page 3 it says: "In general, regression models for non-stationary variables give spurious results. Only exception is if the ...
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### Prime number intercept

Suppose I arrange my (infinite) list of prime numbers in the following way: \begin{array}{c|c}x_i&2&5&11&17&23&31&\cdots\\\hline y_i&3&7&13&19&29&37&...
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### Cook's Distance

I have a problem with calculating Cook Distance (I'm trying to understand it). Ok so here is the task and my 'solution'. I'm asking for comment, is it ok, or what do I wrong. We have simple linear ...
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### Linear Regression's Expectation of Prediction Error on a given point in the test set

I'm self-learning the book "The Elements of Statistical Learning", and I've got a little problem on deriving equation 2.27 in this book. I would appreciate if anyone could help me with this. In order ...
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### What does this absolute value notation mean?

In a regression model function, What does this absolute value notation mean?
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### Minimum-volume confidence ellipsoid for regression with nuisance parameters

I have a linear regression problem with Gaussian errors, nuisance variables and the parameter of interest, $\alpha$. I want to find the smallest-volume region containing the true value of $\alpha$ ...
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### Derivation for marginal effect sizes' distribution in linear model

Model: $$Y_{N\times 1} = X_{N\times M} \beta_{M\times 1} + \epsilon_{N\times 1}$$ the design matrix $X_{N\times M}$ is known, each column of the design matrix has been standardized to have mean 0 and ...
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### Ols estimator with the errors following a bernoulli distribution

I am having trouble understanding how i should approach the following problem: Given 𝑦𝑖 = 𝛼 + 𝛽𝑥𝑖 + 𝜀𝑖 𝑖 = 1, … , N with 𝜀𝑖 𝑖 = 1,2 … , N being a succession of IID Bernoulli ...
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### What is the formula of the linear regression with an error propagation

I am in Physics Licenciature and a day the teacher showed me a formula for the linear regression with error propagation, and time after, I was searching this formula and I didn't find it. Then I am ...
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### What's the Most Appropriate Type of Regression for this Problem?

I have a data set from two groups: firms that use AI and their costs and firms that don't use AI and their costs. Within both groups I have data about their specific costs, e.g. fixed costs, variable ...
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### Linear regression with dependent variables: express prediction with dot products

When dealing with Linear regressin with dependent variables, one can consider the optimization problem: $$\arg \min_w 0.5\lVert{w}\rVert^2 \\ s.t. Xw=y$$ Where $X\in\mathbb{R}^{n,d}$ is the data ...
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### ESL - Linear Model where output is a vector. Ch 2 Pg 11

In the terminology used in ESL, a vector is a column vector. Let output be a $k$-vector, i.e.$$Y=(Y_1,Y_2,\cdots,Y_K)^T$$Now please refer to following line on pg 12. In general $\hat{Y}$ can be a $K$-...
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### Linear regression analysis where the fit values must be greater than the observed values?

Long story short, I would like to efficiently:Minimize ||bX-y||2 subject to X ≥ 0 and bX ≥ y I have an observation that is a single curve (y) in the form of signal intensity vs. frequency. I ...
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### Why residuals are a good estimator of random disturbance?

Let a linear OLS model: $$Y= X \beta + u$$ Where $u$ is a random disturbance. If we define the residual of the regression as $$e = Y - X \widehat{\beta}$$ where $\widehat{\beta}$ is the OLS vector of ...
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### Standard Error Estimate for Beta Coefficients

Suppose I have a linear regression model consisting of $\beta$ estimates, relative to a reference term. Each of these $\beta$s has a $\bar{x}$, a $s_x$, and then $n$, with a calculated $\hat{\sigma}$ ...
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### basic, clear Book about linear regression with examples?

i'm taking a course about linear models , and the book used is Ravishanker "a first course of linear models", the problem is that the book its so theoretical and my background is so basic, so is very ...
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### how affect the variance and the expected value add a variable in a linear model

if the correct model is $Y=X_{1}\beta_{1} + X_{2}\beta_{2} + \varepsilon$, with variance $Var(Y)=\sigma^{2}I$.how change the variance and the expected value of $\beta^{*}_{1}$, in \$Y=X_{1}\beta^{*}_{1}...